Four products are processed sequentially on three machines. The following table gives the pertinent data of the problem.
Manufacturing time (hr) per unit
Machine
Cost per hr ($)
Product 1
Product 2
Product 3
Product4
Capacity (hr)
1
10
2
3
4
2
500
2
5
3
2
1
2
380
3
4
7
3
2
1
450
Unit selling price ($)
75
70
55
45
a) Formulate the problem as an Linear programming model, and find the optimum solution.
Expert Answer
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To formulate the problem as a linear programming model, we need to define our decision variables, objective function, and constraints.
Decision Variables:
Let X1, X2, X3, and X4 be the number of units of product 1, product 2, product 3, and product 4 produced, respectively.
Objective Function:
Maximize profit, which is calculated as the difference between total revenue and total cost. Total revenue is the sum of selling prices of each product multiplied by the number of units sold. Total cost is the sum of machine costs for each product multiplied by the time taken to manufacture each product.
Objective Function: Maximize Z = (75X1 + 70X2 + 55X3 + 45X4) - ((10X1 + 2X2 + 3X3 + 4X4) * 500 + (2X1 + 5X2 + 3X3 + 2X4) * 380 + (4X1 + 7X2 + 3X3 + 2X4) * 450)
Constraints:
Machine 1 Capacity Constraint: The total time taken by all products processed on machine 1 should not exceed the available capacity.
10X1 + 2X2 + 3X3 + 4X4 <= 500
Machine 2 Capacity Constraint: The total time taken by all products processed on machine 2 should not exceed the available capacity.
2X1 + 5X2 + 3X3 + 2X4 <= 380
Machine 3 Capacity Constraint: The total time taken by all products processed on machine 3 should not exceed the available capacity.
4X1 + 7X2 + 3X3 + 2X4 <= 450
Non-negativity Constraint: The number of units produced cannot be negative.
X1 >= 0
X2 >= 0
X3 >= 0
X4 >= 0
Therefore, the linear programming model can be formulated as:
Maximize Z = (75X1 + 70X2 + 55X3 + 45X4) - ((10X1 + 2X2 + 3X3 + 4X4) * 500 + (2X1 + 5X2 + 3X3 + 2X4) * 380 + (4X1 + 7X2 + 3X3 + 2X4) * 450)
Subject to:
10X1 + 2X2 + 3X3 + 4X4 <= 500
2X1 + 5X2 + 3X3 + 2X4 <= 380
4X1 + 7X2 + 3X3 + 2X4 <= 450
X1 >= 0, X2 >= 0, X3 >= 0, X4 >= 0
To find the optimum solution, we can solve this linear programming model using a solver or optimization software.
To formulate the problem as a linear programming model, we need to define decision variables, objective function, and constraints.
Let's define the decision variables:
Let x1, x2, x3, and x4 represent the number of units of products 1, 2, 3, and 4, respectively, to be produced.
Objective function:
We want to maximize the profit. The profit can be calculated as the selling price per unit multiplied by the number of units produced, minus the manufacturing cost per unit multiplied by the number of units produced. Therefore, the objective function is:
Maximize: 75x1 + 70x2 + 55x3 + 45x4 - (10x1 + 2x2 + 3x3 + 4x4) - (2x2 + 5x3 + 3x4) - (2x1 + x2 + 2x3)
Constraints:
1. Machine 1's capacity should not be exceeded:
10x1 + 2x2 + 3x3 + 4x4 <= 500
2. Machine 2's capacity should not be exceeded:
2x2 + 5x3 + 3x4 <= 380
3. Machine 3's capacity should not be exceeded:
2x1 + x2 + 2x3 <= 450
4. Non-negativity constraints:
x1 >= 0, x2 >= 0, x3 >= 0, x4 >= 0
The linear programming model can be summarized as follows:
Maximize: 75x1 + 70x2 + 55x3 + 45x4 - 10x1 - 2x2 - 3x3 - 4x4 - 2x2 - 5x3 - 3x4 - 2x1 - x2 - 2x3
Subject to:
10x1 + 2x2 + 3x3 + 4x4 <= 500
2x2 + 5x3 + 3x4 <= 380
2x1 + x2 + 2x3 <= 450
x1 >= 0, x2 >= 0, x3 >= 0, x4 >= 0
Using a linear programming solver, we can solve this model to find the optimum solution.
To formulate this problem as a linear programming model, we need to define decision variables, the objective function, and the constraints.
Decision Variables:
Let x1, x2, x3, and x4 represent the number of units of Product 1, Product 2, Product 3, and Product 4, respectively, to be produced.
Objective Function:
The objective function is to maximize the total profit obtained from selling these products. The total profit can be calculated as the sum of the profits made from each product:
Maximize Z = 75x1 + 70x2 + 55x3 + 45x4
Constraints:
1. Machine 1 capacity constraint:
The total time required to process all products on Machine 1 cannot exceed its capacity of 500 hours:
10x1 + 2x2 + 3x3 + 4x4 <= 500
2. Machine 2 capacity constraint:
The total time required to process all products on Machine 2 cannot exceed its capacity of 380 hours:
2x1 + 5x2 + 3x3 + 2x4 <= 380
3. Machine 3 capacity constraint:
The total time required to process all products on Machine 3 cannot exceed its capacity of 450 hours:
4x1 + 7x2 + 3x3 + 2x4 <= 450
Non-negativity constraint:
The number of units of each product cannot be negative:
x1, x2, x3, x4 >= 0
Now that we have formulated the problem, we can solve it using a linear programming solver to find the optimum solution.